changebasis sometimes returns incorrect nonzero element

[hyrodium]

julia> using BasicBSpline

julia> p = 3
3

julia> P1 = BSplineSpace{p}(KnotVector(1:8))
BSplineSpace{3, Float64}(KnotVector([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]))

julia> P2 = BSplineSpace{p}(KnotVector(2,3,3,4,5,6,7,8))
BSplineSpace{3, Float64}(KnotVector([2.0, 3.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]))

julia> changebasis(P1,P2)
4×4 Matrix{Float64}:
  0.666667      2.04041e-16  -1.41585e-16  3.30839e-16
  0.333333      1.0          -1.18601e-15  5.25506e-15
 -1.73591e-15   9.12098e-16   1.0          4.36688e-15
  9.68114e-16  -3.80046e-16   3.74332e-16  1.0

[hyrodium]

Note that there still be some incorrect non-zero with changebasis function.

julia> using BasicBSpline

julia> p = 3
3

julia> P1 = BSplineSpace{p}(KnotVector(1:8))
BSplineSpace{3, Int64}(KnotVector([1, 2, 3, 4, 5, 6, 7, 8]))

julia> P2 = BSplineSpace{p}(KnotVector(2,3,3,4,5,6,7,8))
BSplineSpace{3, Int64}(KnotVector([2, 3, 3, 4, 5, 6, 7, 8]))

julia> BasicBSpline._changebasis_sim(P1,P2)
4×4 Matrix{Float64}:
 0.666667  0.0  0.0  0.0
 0.333333  1.0  0.0  0.0
 0.0       0.0  1.0  0.0
 0.0       0.0  0.0  1.0

julia> changebasis(P1,P2)
4×4 Matrix{Float64}:
 0.666667     0.0  0.0   0.0
 0.333333     1.0  0.0  -5.55112e-17
 3.70074e-17  0.0  1.0  -2.22045e-16
 0.0          0.0  0.0   1.0

[hyrodium]

Here's another case:

julia> p = 3
3

julia> P = BSplineSpace{p}(KnotVector(1:10))
BSplineSpace{3, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]))

julia> P′ = BSplineSpace{p}(KnotVector([1,2,3,4,4.5,5,6,7,8.1,9.1,10.5]))
BSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([1.0, 2.0, 3.0, 4.0, 4.5, 5.0, 6.0, 7.0, 8.1, 9.1, 10.5]))

julia> changebasis(P,P′)
6×7 Matrix{Float64}:
  1.0           0.166667     0.0          0.0       0.0           0.0           0.0
  8.88178e-16   0.833333     0.5          0.0       0.0           0.0           0.0
  1.33227e-15  -5.55112e-17  0.5          0.833333  5.55112e-17   5.55112e-17  -3.05866e-16
 -5.82867e-16   4.16334e-17  4.16334e-17  0.166667  1.0          -0.0333333     0.0183333
  0.0           0.0          0.0          0.0       0.0           1.03333      -0.103333
  0.0           0.0          0.0          0.0       0.0           0.0           1.085